Some Tools for Exploring Supersymmetric RG Flows

Transcription

1 Some Tools for Exploring Supersymmetric RG Flows Thomas Dumitrescu Harvard University Work in Progress with G. Festuccia and M. del Zotto NatiFest, September 2016 IAS, Princeton

2 Quantum Field Theory and Supersymmetry Despite its phenomenal successes, QFT is still a work in progress: There are indications that we are still missing big things perhaps quantum field theory should be reformulated Nathan Seiberg 1

3 Quantum Field Theory and Supersymmetry Despite its phenomenal successes, QFT is still a work in progress: There are indications that we are still missing big things perhaps quantum field theory should be reformulated Nathan Seiberg Attempts to do better are limited by our (in-) ability to control the dynamics of non-trivial, interacting field theories. Natural to look for simplifying limits, with additional symmetries: topological, conformal, weak-coupling, large-n, integrable,... 1

4 Quantum Field Theory and Supersymmetry Despite its phenomenal successes, QFT is still a work in progress: There are indications that we are still missing big things perhaps quantum field theory should be reformulated Nathan Seiberg Attempts to do better are limited by our (in-) ability to control the dynamics of non-trivial, interacting field theories. Natural to look for simplifying limits, with additional symmetries: topological, conformal, weak-coupling, large-n, integrable,... Supersymmetric QFTs can display rich, non-conformal dynamics. They also have some protected (BPS) quantities that can be analyzed exactly. Example: superpotential W (Φ), holomorphic in all (background) chiral superfields, including couplings [Seiberg]. In favorable situations, tracking the protected quantities along the RG flow can give a good picture of the dynamics. 1

5 Supersymmetric Indices With this in mind, we would like to expand our toolbox of protected observables, and to deepen our understanding of them. 2

6 Supersymmetric Indices With this in mind, we would like to expand our toolbox of protected observables, and to deepen our understanding of them. A large and interesting class: SUSY partition functions Z M on a compact spacetime manifold M. Example [Witten]: M = T d, Z M = Tr H ( 1) F, H = states on T d 1 R time 2

7 Supersymmetric Indices With this in mind, we would like to expand our toolbox of protected observables, and to deepen our understanding of them. A large and interesting class: SUSY partition functions Z M on a compact spacetime manifold M. Example [Witten]: M = T d, Z M = Tr H ( 1) F, H = states on T d 1 R time Naturally defined in any (non-conformal) SUSY theory. Counts (with sign) the SUSY vacua on T d 1 : index. Varying parameters (RG flow) typically does not change the answer, but vacua can pair up and acquire positive energy. Subtle wall-crossing phenomena, sometimes ill defined. 2

8 Supersymmetric Indices (cont.) Recently, many examples of supersymmetric indices defined as partition functions on manifolds of topology M = S d 1 S 1 : Count supersymmetric states on S d 1 R time. Naturally defined in SCFTs (via a conformal map). 3

9 Supersymmetric Indices (cont.) Recently, many examples of supersymmetric indices defined as partition functions on manifolds of topology M = S d 1 S 1 : Count supersymmetric states on S d 1 R time. Naturally defined in SCFTs (via a conformal map). Count BPS local operators in flat space (state-operator correspondence) [Kinney-Maldacena-Minwalla-Raju]. Independent of exactly marginal couplings. Robust due to discrete spectrum and normalizable vacuum on S d 1. 3

10 Supersymmetric Indices (cont.) Recently, many examples of supersymmetric indices defined as partition functions on manifolds of topology M = S d 1 S 1 : Count supersymmetric states on S d 1 R time. Naturally defined in SCFTs (via a conformal map). Count BPS local operators in flat space (state-operator correspondence) [Kinney-Maldacena-Minwalla-Raju]. Independent of exactly marginal couplings. Robust due to discrete spectrum and normalizable vacuum on S d 1. Just as the index on M = T d, indices on M = S d 1 S 1 can sometimes be defined for non-conformal supersymmetric theories: When can this be done, how to preserve SUSY (not obvious)? Not canonical: additional choices, parameters in curved space. Does the index depend on them? What does it count? Only non-conformal indices can be used to explore RG flows. 3

11 SUSY QFT in Curved Space A systematic approach to supersymmetric QFT in curved space was developed by [Festuccia-Seiberg] 4

12 SUSY QFT in Curved Space A systematic approach to supersymmetric QFT in curved space was developed by [Festuccia-Seiberg] for today [Nati-Fest(uccia)]. 4

13 SUSY QFT in Curved Space A systematic approach to supersymmetric QFT in curved space was developed by [Festuccia-Seiberg] for today [Nati-Fest(uccia)]. Main Idea: the non-dynamical metric g µν on M must reside in an off-shell supergravity multiplet. This extends the powerful principle that all background fields should reside in superfields [Seiberg]. This formalism was recently reviewed in arxiv: [TD]. 4

14 SUSY QFT in Curved Space A systematic approach to supersymmetric QFT in curved space was developed by [Festuccia-Seiberg] for today [Nati-Fest(uccia)]. Main Idea: the non-dynamical metric g µν on M must reside in an off-shell supergravity multiplet. This extends the powerful principle that all background fields should reside in superfields [Seiberg]. This formalism was recently reviewed in arxiv: [TD]. The coupling of the QFT to background supergravity proceeds via the flat-space stress tensor T µν, and its superpartners J i B and J i F, L = 1 2 gµν T µν + i ( ) BBJ i B i + BF i JF i + (seagull terms) 4

15 SUSY QFT in Curved Space (cont.) L = 1 2 gµν T µν + i ( ) BBJ i B i + BF i JF i + (seagull terms) Typically, activate bosons g µν, BB i and set fermions BF i = 0. A supercharge Q exists if δ Q BF i = 0 µ ζ + f ( g µν, BB) i ζ = 0. These equations determine all SUSY backgrounds (g µν, BB). i 5

16 SUSY QFT in Curved Space (cont.) L = 1 2 gµν T µν + i ( ) BBJ i B i + BF i JF i + (seagull terms) Typically, activate bosons g µν, B i B and set fermions B i F = 0. A supercharge Q exists if δ Q B i F = 0 µ ζ + f ( g µν, B i B) ζ = 0. These equations determine all SUSY backgrounds (g µν, B i B). Activating only g µν is typically not enough to preserve SUSY ([Q, T µν ] 0), or to specify the background (different B i B). SUSY algebra, Lagrangians on M follow from supergravity. 5

17 An N = 1 Index in 4d There are unitary N = 1 theories on Sl 3 R time [Sen, Römelsberger, Festuccia-Seiberg]. The SUSY algebra is deformed to su(2 1): } {Q α, Q β = δ α β (H + 1 ) l R + 2 l J α { } β, Qα, Q β = 0 su(2) u(1) subalgebra = (left or right SU(2) isometries of S 3 ) (unbroken U(1) R -symmetry). The Hamiltonian H is central. 6

18 An N = 1 Index in 4d There are unitary N = 1 theories on Sl 3 R time [Sen, Römelsberger, Festuccia-Seiberg]. The SUSY algebra is deformed to su(2 1): } {Q α, Q β = δ α β (H + 1 ) l R + 2 l J α { } β, Qα, Q β = 0 su(2) u(1) subalgebra = (left or right SU(2) isometries of S 3 ) (unbroken U(1) R -symmetry). The Hamiltonian H is central. L A µ j (R) µ + V µ X µ, A 0 V 0 1 l 6

19 An N = 1 Index in 4d There are unitary N = 1 theories on Sl 3 R time [Sen, Römelsberger, Festuccia-Seiberg]. The SUSY algebra is deformed to su(2 1): } {Q α, Q β = δ α β (H + 1 ) l R + 2 l J α { } β, Qα, Q β = 0 su(2) u(1) subalgebra = (left or right SU(2) isometries of S 3 ) (unbroken U(1) R -symmetry). The Hamiltonian H is central. L A µ j (R) µ + V µ X µ, A 0 V 0 1 l We can choose any U(1) R current j (R) µ in flat space. The couplings on S 3 explicitly depend on this choice. The coupling V µ X µ only exists in non-conformal theories; it is crucial for preserving supersymmetry. In an SCFT: su(2 1) superconformal algebra, with Q α S α and H D R 6

20 An N = 1 Index in 4d (cont.) su(2 1) unitarity bounds: El 2j + 2 r or El = r (j = 0). Count short multiplets (modulo recombination) using an index: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l 7

21 An N = 1 Index in 4d (cont.) su(2 1) unitarity bounds: El 2j + 2 r or El = r (j = 0). Count short multiplets (modulo recombination) using an index: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l Non-renormalization theorem [Festuccia-Seiberg]: I(q) is independent of all deformations preserving su(2 1), because the energy of all short representations is fixed in terms of j, r. 7

22 An N = 1 Index in 4d (cont.) su(2 1) unitarity bounds: El 2j + 2 r or El = r (j = 0). Count short multiplets (modulo recombination) using an index: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l Non-renormalization theorem [Festuccia-Seiberg]: I(q) is independent of all deformations preserving su(2 1), because the energy of all short representations is fixed in terms of j, r. Set all couplings to zero, compute in a free SCFT: simple matrix integral counting gauge-invariant local operators. 7

23 An N = 1 Index in 4d (cont.) su(2 1) unitarity bounds: El 2j + 2 r or El = r (j = 0). Count short multiplets (modulo recombination) using an index: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l Non-renormalization theorem [Festuccia-Seiberg]: I(q) is independent of all deformations preserving su(2 1), because the energy of all short representations is fixed in terms of j, r. Set all couplings to zero, compute in a free SCFT: simple matrix integral counting gauge-invariant local operators. The RG flow itself is an allowed deformation I(q) can be computed anywhere along the flow. It must match across IR (or [Seiberg]) dualities [Römelsberger,Dolan-Osborn]. 7

24 An N = 1 Index in 4d (cont.) su(2 1) unitarity bounds: El 2j + 2 r or El = r (j = 0). Count short multiplets (modulo recombination) using an index: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l Non-renormalization theorem [Festuccia-Seiberg]: I(q) is independent of all deformations preserving su(2 1), because the energy of all short representations is fixed in terms of j, r. Set all couplings to zero, compute in a free SCFT: simple matrix integral counting gauge-invariant local operators. The RG flow itself is an allowed deformation I(q) can be computed anywhere along the flow. It must match across IR (or [Seiberg]) dualities [Römelsberger,Dolan-Osborn]. Everything is well defined as long as the spectrum of H is discrete. 7

25 Is N = 2 Harder Than N = 1? When the R-charge r of a scalar φ vanishes, the spectrum of H is continuous, because the r-dependent curvature coupling vanishes: L µ φ µ φ + f(r) l 2 φ 2, f(r = 0) = 0 The flat direction implies a divergence in the index I(q) = Z S 3 l S 1. 8

26 Is N = 2 Harder Than N = 1? When the R-charge r of a scalar φ vanishes, the spectrum of H is continuous, because the r-dependent curvature coupling vanishes: L µ φ µ φ + f(r) l 2 φ 2, f(r = 0) = 0 The flat direction implies a divergence in the index I(q) = Z S 3 l S 1. This problem naturally arises in N = 2 theories: N = 2 SCFTs have SU(2) R U(1) R symmetry, which can be used to define a well-behaved index [Kinney et. al.]. Non-conformal N = 2 theories often preserve the SU(2) R symmetry, but typically not the U(1) R symmetry. Vector-multiplet scalars φ are neutral under SU(2) R, i.e. I(q) does not exist in non-conformal N = 2 theories with vectors. We expect that N = 2 supersymmetry allows us to do better! 8

27 Searching for a Non-Conformal N = 2 Index As a guide, we start with an SCFT, but avoid certain generators: Conformally map the theory to S 3 l R. Look for a subalgebra of the superconformal algebra that only includes genuine isometries of S 3 l R and SU(2) R. 9

28 Searching for a Non-Conformal N = 2 Index As a guide, we start with an SCFT, but avoid certain generators: Conformally map the theory to S 3 l R. Look for a subalgebra of the superconformal algebra that only includes genuine isometries of S 3 l R and SU(2) R. The largest such subalgebra is another, but different su(2 1): } ( {Q α, Q β = δ α β H + 1 ) l R l J α β, R 3 = Cartan generator of the SU(2) R symmetry. { Qα, Q β } = 0 Key: J α β generates the diagonal SU(2) isometries of S 3 l. 9

29 Searching for a Non-Conformal N = 2 Index As a guide, we start with an SCFT, but avoid certain generators: Conformally map the theory to S 3 l R. Look for a subalgebra of the superconformal algebra that only includes genuine isometries of S 3 l R and SU(2) R. The largest such subalgebra is another, but different su(2 1): } ( {Q α, Q β = δ α β H + 1 ) l R l J α β, R 3 = Cartan generator of the SU(2) R symmetry. { Qα, Q β } = 0 Key: J α β generates the diagonal SU(2) isometries of S 3 l. Short su(2 1) superconformal multiplets satisfy = j diag. + 2R 3. The su(2 1) index I(q) is the Schur limit of the N = 2 superconformal index [Gadde-Rastelli-Razamat-Yan]. 9

30 Supergravity Background We would like to construct non-conformal N = 2 theories on S 3 R time that realize this diagonal su(2 1) SUSY algebra. 10

31 Supergravity Background We would like to construct non-conformal N = 2 theories on S 3 R time that realize this diagonal su(2 1) SUSY algebra. Use background supergravity formalism of [Festuccia-Seiberg]: 1.) Choose a stress-tensor multiplet in flat space. Essentially all interesting N = 2 theories with an SU(2) R symmetry have a distinguished stress-tensor multiplet discovered by [Sohnius]: T ψ i α W [µν], R (ij) µ, r µ S i µα T µν (real) vanishes in SCFT: X (ij) χ i α z µ, K (complex) 10

32 Supergravity Background We would like to construct non-conformal N = 2 theories on S 3 R time that realize this diagonal su(2 1) SUSY algebra. Use background supergravity formalism of [Festuccia-Seiberg]: 1.) Choose a stress-tensor multiplet in flat space. Essentially all interesting N = 2 theories with an SU(2) R symmetry have a distinguished stress-tensor multiplet discovered by [Sohnius]: T ψ i α W [µν], R (ij) µ, r µ S i µα T µν (real) vanishes in SCFT: X (ij) χ i α z µ, K (complex) 2.) Need a background supergravity field for each operator: L = J T T + J [µν] W W [µν] + V µ(ij) R µ(ij) + A µ r µ 1 2 gµν T µν + J (ij) X X (ij) + C µ z µ + J K K + (c.c.) + (fermions) Note: C µ is the dimensionless central-charge gauge field. 10

33 Supergravity Background (cont.) 3.) Solve the SUSY conditions δ Q (SUGRA fermions) = 0. We found a solution with a round metric on S 3 l of radius l, ds 2 = dt 2 + l 2 ( dθ 2 + sin 2 θdω 2 ), 0 θ π 11

34 Supergravity Background (cont.) 3.) Solve the SUSY conditions δ Q (SUGRA fermions) = 0. We found a solution with a round metric on S 3 l of radius l, ds 2 = dt 2 + l 2 ( dθ 2 + sin 2 θdω 2 ), 0 θ π sz :?_ 2 0=0 IT I %, co 11

35 Supergravity Background (cont.) 3.) Solve the SUSY conditions δ Q (SUGRA fermions) = 0. We found a solution with a round metric on S 3 l of radius l, ds 2 = dt 2 + l 2 ( dθ 2 + sin 2 θdω 2 ), 0 θ π sz :?_ 2 0=0 IT I %, co Some other fields break SO(4) SU(2) diag, SU(2) R R 3, J 3 X ζ l cos θ, C µdx µ ζ cos θdt (decouple in SCFT) J T 1 l 2, V 3 µ dx µ 1 l dt, J K = ζ 2 = constant phase 11

36 Supergravity Background (cont.) In fact, there are four supercharges that give the desired su(2 1). We can analyze N = 2 theories on this S 3 R time background: 12

37 Supergravity Background (cont.) In fact, there are four supercharges that give the desired su(2 1). We can analyze N = 2 theories on this S 3 R time background: The theory is unitary standard reality for background fields. 12

38 Supergravity Background (cont.) In fact, there are four supercharges that give the desired su(2 1). We can analyze N = 2 theories on this S 3 R time background: The theory is unitary standard reality for background fields. No problem with vector multiplets: the background induces a mass term for the scalars φ that lifts the flat direction. 12

39 Supergravity Background (cont.) In fact, there are four supercharges that give the desired su(2 1). We can analyze N = 2 theories on this S 3 R time background: The theory is unitary standard reality for background fields. No problem with vector multiplets: the background induces a mass term for the scalars φ that lifts the flat direction. In an SCFT, all terms that break SO(4), SU(2) R can be removed to recover a conformally coupled theory. In that case, the phase ζ specifies the embedding of su(2 1) into the superconformal algebra. 12

40 Supergravity Background (cont.) In fact, there are four supercharges that give the desired su(2 1). We can analyze N = 2 theories on this S 3 R time background: The theory is unitary standard reality for background fields. No problem with vector multiplets: the background induces a mass term for the scalars φ that lifts the flat direction. In an SCFT, all terms that break SO(4), SU(2) R can be removed to recover a conformally coupled theory. In that case, the phase ζ specifies the embedding of su(2 1) into the superconformal algebra. In non-conformal theories, the background fields J 3 X, C 0 that break SO(4), SU(2) R lead to position-dependent mass terms and derivative couplings. 12

41 Why is the Matrix Model Correct? The position-dependent couplings look dauting! But we can apply the su(2 1) non-renormalization theorem of [Festuccia-Seiberg]: 13

42 Why is the Matrix Model Correct? The position-dependent couplings look dauting! But we can apply the su(2 1) non-renormalization theorem of [Festuccia-Seiberg]: The index is independent of all continuous couplings: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l 13

43 Why is the Matrix Model Correct? The position-dependent couplings look dauting! But we can apply the su(2 1) non-renormalization theorem of [Festuccia-Seiberg]: The index is independent of all continuous couplings: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l In any renormalizable gauge theory with matter: set gauge couplings, masses to zero compute in the free UV theory. This leads to exactly the same matrix model as in conformal gauge theories [Gadde-Rastelli-Razamat-Yan]. The integrand is minimally modified to reflect the non-conformal matter [...]. 13

44 Why is the Matrix Model Correct? The position-dependent couplings look dauting! But we can apply the su(2 1) non-renormalization theorem of [Festuccia-Seiberg]: The index is independent of all continuous couplings: I(q) = Tr H ( 1) F q H = Z S 3 l S 1(q), log q radius(s1 ) l In any renormalizable gauge theory with matter: set gauge couplings, masses to zero compute in the free UV theory. This leads to exactly the same matrix model as in conformal gauge theories [Gadde-Rastelli-Razamat-Yan]. The integrand is minimally modified to reflect the non-conformal matter [...]. Also possible to compute I(q) in the IR make contact with recent conjectures of [Iqbal-Vafa, Cordova-Shao + Gaiotto,...] relating I(q) to BPS particles on the Coulomb branch. 13

45 Comments on (Non-) Decoupling We argued that I(q) does not depend on continuous parameters, e.g. a mass m for a free hypermultiplet. 14

46 Comments on (Non-) Decoupling We argued that I(q) does not depend on continuous parameters, e.g. a mass m for a free hypermultiplet. This is inconsistent with naive decoupling: as m, the flat-space theory becomes trivial, with I(q) = 1. On S 3 l R time, we expect non-vacuum states to have energy E m. 14

47 Comments on (Non-) Decoupling (cont.) In fact, the position-dependent background fields lead to a non-trivial mass function m 2 (θ) for some scalar modes, e.g. mzco ) ~m2t- }n±yt =o e to * Near the poles, m 2 (θ) can become very small. This leads to localized modes with E 1 l m, which do not decouple. In natural units, the background fields are very strong: m 2 (θ) m 2 C0 2 Cphys. 2, C 0 cos θ, 15

48 Inserting BPS Line Operators At the poles of S 3 the S 2 shrinks, su(2 1) contracts to a subalgebra of flat-space SUSY. It is the algebra preserved by a BPS particle with central charge Z ζ (or its antiparticle). 16

49 Inserting BPS Line Operators At the poles of S 3 the S 2 shrinks, su(2 1) contracts to a subalgebra of flat-space SUSY. It is the algebra preserved by a BPS particle with central charge Z ζ (or its antiparticle). This algebra is also preserved by certain 1 2-BPS line operators, studied by [Gaiotto-Moore-Neitzke,...]. They are: Supported on straight lines L. Invariant under the maximal unbroken SU(2) R SU(2) rot. 16

50 Inserting BPS Line Operators At the poles of S 3 the S 2 shrinks, su(2 1) contracts to a subalgebra of flat-space SUSY. It is the algebra preserved by a BPS particle with central charge Z ζ (or its antiparticle). This algebra is also preserved by certain 1 2-BPS line operators, studied by [Gaiotto-Moore-Neitzke,...]. They are: Supported on straight lines L. Invariant under the maximal unbroken SU(2) R SU(2) rot. Example: a Wilson line of charge q for a U(1) gauge field A, ( W q = exp iq (A + i L 2ζ φ iζ ) 2 φ) These line defects can be inserted into our S 3 l R time background, if we place them at the poles and along time. 16

51 Deforming the Sphere The background admits a family of deformations where the radius of S 2 is any bounded function f(θ) on the interval 0 θ π, ds 2 = dt 2 + l 2 ( dθ 2 + f(θ) 2 dω 2 ) ' ooe I s h * R 17

52 Deforming the Sphere The background admits a family of deformations where the radius of S 2 is any bounded function f(θ) on the interval 0 θ π, ds 2 = dt 2 + l 2 ( dθ 2 + f(θ) 2 dω 2 ) ' ooe I s h * R Any choice of f(θ) preserves the full su(2 1) symmetry again the indx I(q) is unchanged. If f(θ) = const. the geometry is S 2 R 1,1 and the SUSY algebra enhances to su(2 2). Theories with this algebra were studied by [Itzhaki-Kutasov-Seiberg, Lin-Maldacena,...]. 17

53 A State-Operator Map for BPS Lines The unitarity bounds of su(2 2) show that the theory on S 2 R 1,1 must be fully gapped. It can have multiple isolated vacua that are invariant under SU(2) R SU(2) rot.. They cannot be lifted by any continuous parameter variations, including RG flow (very rigid). 18

54 A State-Operator Map for BPS Lines The unitarity bounds of su(2 2) show that the theory on S 2 R 1,1 must be fully gapped. It can have multiple isolated vacua that are invariant under SU(2) R SU(2) rot.. They cannot be lifted by any continuous parameter variations, including RG flow (very rigid). Claim: Vacua 1 2-BPS line defects. This follows from path integrals on a semi-infinite cigar similar to tt in 2d [Cecotti-Vafa]. 18

55 A State-Operator Map for BPS Lines The unitarity bounds of su(2 2) show that the theory on S 2 R 1,1 must be fully gapped. It can have multiple isolated vacua that are invariant under SU(2) R SU(2) rot.. They cannot be lifted by any continuous parameter variations, including RG flow (very rigid). Claim: Vacua 1 2-BPS line defects. This follows from path integrals on a semi-infinite cigar similar to tt in 2d [Cecotti-Vafa]. Example: free U(1) gauge theory on S 2 R 1,1 leads to axion electrodynamics on R 1,1, with vacua labeled by any integer q Z: L 2d F ( ϕ) 2 + ϕf 01, ϕ = (const.) q These vacua are 1 2 -BPS Wilson lines W q of charge q. Another copy of L 2d leads to vacua corresponding to t Hooft lines. 18

56 A State-Operator Map for BPS Lines The unitarity bounds of su(2 2) show that the theory on S 2 R 1,1 must be fully gapped. It can have multiple isolated vacua that are invariant under SU(2) R SU(2) rot.. They cannot be lifted by any continuous parameter variations, including RG flow (very rigid). Claim: Vacua 1 2-BPS line defects. This follows from path integrals on a semi-infinite cigar similar to tt in 2d [Cecotti-Vafa]. Example: free U(1) gauge theory on S 2 R 1,1 leads to axion electrodynamics on R 1,1, with vacua labeled by any integer q Z: L 2d F ( ϕ) 2 + ϕf 01, ϕ = (const.) q These vacua are 1 2 -BPS Wilson lines W q of charge q. Another copy of L 2d leads to vacua corresponding to t Hooft lines. The correspondence explains many observed features of these BPS defects, e.g. the one-to-one map between UV lines and IR lines on the Coulomb branch [Gaiotto-Moore-Neitzke, Cordova-Neitzke,...]. 18

57 Conclusions We defined a new S 3 S 1 index I(q) for non-conformal N = 2 theories generalizes the superconformal Schur index. In both cases, supergravity background fields and an su(2 1) non-renormalization theorem played a crucial role. In asymptotically free or conformal gauge theories, I(q) can be computed using a simple matrix model (UV). I(q) is independent of mass deformations naively violates decoupling of heavy states. No paradox: the background fields are very strong and can make some massive states light. Goal: compute I(q) in the IR, or perhaps some intermediate description that includes massive (BPS) particles. I(q) can be decorated with 1 2-BPS line defects. They are in one-to-one correspondence with the massive vacua of the theory on an S 2 R 1,1 background with su(2 2) symmetry. 19

58 Thank You for Your Attention and Happy Birthday Nati! 20